Noncommutative boundaries and hyperrigidity of operator systems

Classical Shilov and Choquet boundaries provide analogues of the maximal modulus principle and peaking phenomenon for abstract function systems. Arveson extended this to noncommutative boundary theory by defining Shilov and Choquet boundaries for operator systems within $C^*$-algebras, offering crucial tools for their study. He demonstrated that many results from commutative boundary theory have noncommutative analogs. Among his later works, in an attempt to generalize approximation rigidity results for function systems, Arveson introduced the concept of hyperrigidity for operator systems and conjectured that an operator system is hyperrigid if and only if its Choquet boundary is everything. However, recent work by the author and Dor-On has produced a counterexample to this conjecture, revealing new challenges in defining noncommutative rigidity for approximations and understanding the gap between the Choquet boundary being everything and hyperrigidity.
In this talk, I will outline the principles of noncommutative boundary theory and demonstrate the tools provided by dilation theory. I will present Arveson's hyperrigidity conjecture, explore the construction of counterexamples, and show how operator systems associated with correspondences offer a natural framework for discovering new counterexamples and testing emerging conjectures. In particular, I will demonstrate that the conjecture holds for certain well-known and broad classes of $C^*$-correspondences.